Linear Algebra for Beginners

          Module 1: Introduction to Linear Algebra Fundamentals

Section 1: Overview of Linear Algebra  
- Lesson 1: What is Linear Algebra?  
  Module 1, Section 1, Lesson 1 What is Linear Algebra?  
- Lesson 2: The Role of Linear Algebra in Mathematics  
  Module 1, Section 1, Lesson 2 The Role of Linear Algebra in Mathematics  

Section 2: Basic Terminology and Symbols  
- Lesson 1: Understanding Vectors and Scalars  
  Module 1, Section 2, Lesson 1 Understanding Vectors and Scalars  
- Lesson 2: Introduction to Matrices and Notation  
  Module 1, Section 2, Lesson 2 Introduction to Matrices and Notation  

Section 3: History and Development  
- Lesson 1: Early Concepts and Influences  
  Module 1, Section 3, Lesson 1 Early Concepts and Influences  
- Lesson 2: Milestones in Linear Algebra  
  Module 1, Section 3, Lesson 2 Milestones in Linear Algebra  

Section 4: Importance and Applications  
- Lesson 1: Linear Algebra in Science and Engineering  
  Module 1, Section 4, Lesson 1 Linear Algebra in Science and Engineering  
- Lesson 2: Career Paths Involving Linear Algebra  
  Module 1, Section 4, Lesson 2 Career Paths Involving Linear Algebra  

Section 5: Preparing for Problem Solving  
- Lesson 1: Developing a Mathematical Mindset  
  Module 1, Section 5, Lesson 1 Developing a Mathematical Mindset  
- Lesson 2: Strategies for Learning New Concepts  
  Module 1, Section 5, Lesson 2 Strategies for Learning New Concepts  

Module 2: Understanding Vector Spaces

Section 1: Introduction to Vectors  
- Lesson 1: What Are Vectors?  
  Module 2, Section 1, Lesson 1 What Are Vectors?  
- Lesson 2: Visualizing Vectors in Space  
  Module 2, Section 1, Lesson 2 Visualizing Vectors in Space  

Section 2: Vector Operations  
- Lesson 1: Vector Addition and Subtraction  
  Module 2, Section 2, Lesson 1 Vector Addition and Subtraction  
- Lesson 2: Scalar Multiplication  
  Module 2, Section 2, Lesson 2 Scalar Multiplication  

Section 3: Properties of Vector Spaces  
- Lesson 1: Linear Combinations and Span  
  Module 2, Section 3, Lesson 1 Linear Combinations and Span  
- Lesson 2: Basis and Dimension  
  Module 2, Section 3, Lesson 2 Basis and Dimension  

Section 4: Subspaces and Their Properties  
- Lesson 1: Defining Subspaces  
  Module 2, Section 4, Lesson 1 Defining Subspaces  
- Lesson 2: Examples of Subspaces  
  Module 2, Section 4, Lesson 2 Examples of Subspaces  

Section 5: Applications of Vectors  
- Lesson 1: Vectors in Physics and Engineering  
  Module 2, Section 5, Lesson 1 Vectors in Physics and Engineering  
- Lesson 2: Problem Solving with Vector Concepts  
  Module 2, Section 5, Lesson 2 Problem Solving with Vector Concepts  

Module 3: Matrix Operations and Transformations

Section 1: Introduction to Matrices  
- Lesson 1: What is a Matrix?  
  Module 3, Section 1, Lesson 1 What is a Matrix?  
- Lesson 2: Representing Data with Matrices  
  Module 3, Section 1, Lesson 2 Representing Data with Matrices  

Section 2: Basic Matrix Operations  
- Lesson 1: Matrix Addition and Subtraction  
  Module 3, Section 2, Lesson 1 Matrix Addition and Subtraction  
- Lesson 2: Scalar Multiplication of Matrices  
  Module 3, Section 2, Lesson 2 Scalar Multiplication of Matrices  

Section 3: Matrix Multiplication  
- Lesson 1: Rules and Procedures for Multiplying Matrices  
  Module 3, Section 3, Lesson 1 Rules and Procedures for Multiplying Matrices  
- Lesson 2: Understanding the Dot Product Approach  
  Module 3, Section 3, Lesson 2 Understanding the Dot Product Approach  

Section 4: Special Types of Matrices  
- Lesson 1: Identity, Zero, and Diagonal Matrices  
  Module 3, Section 4, Lesson 1 Identity, Zero, and Diagonal Matrices  
- Lesson 2: Symmetric and Transpose Matrices  
  Module 3, Section 4, Lesson 2 Symmetric and Transpose Matrices  

Section 5: Matrix Transformations  
- Lesson 1: Linear Transformations and Their Matrices  
  Module 3, Section 5, Lesson 1 Linear Transformations and Their Matrices  
- Lesson 2: Geometric Interpretations of Transformations  
  Module 3, Section 5, Lesson 2 Geometric Interpretations of Transformations  

Module 4: Systems of Linear Equations

Section 1: Introduction to Systems  
- Lesson 1: What are Systems of Equations?  
  Module 4, Section 1, Lesson 1 What are Systems of Equations?  
- Lesson 2: Representing Systems with Matrices  
  Module 4, Section 1, Lesson 2 Representing Systems with Matrices  

Section 2: Solving Systems: Graphical and Substitution Methods  
- Lesson 1: Graphical Approaches to Systems  
  Module 4, Section 2, Lesson 1 Graphical Approaches to Systems  
- Lesson 2: Substitution and Elimination Techniques  
  Module 4, Section 2, Lesson 2 Substitution and Elimination Techniques  

Section 3: Row Reduction and Echelon Forms  
- Lesson 1: The Process of Gaussian Elimination  
  Module 4, Section 3, Lesson 1 The Process of Gaussian Elimination  
- Lesson 2: Reduced Row Echelon Form (RREF)  
  Module 4, Section 3, Lesson 2 Reduced Row Echelon Form (RREF)  

Section 4: Applications of Systems of Equations  
- Lesson 1: Systems in Real-World Problem Solving  
  Module 4, Section 4, Lesson 1 Systems in Real-World Problem Solving  
- Lesson 2: Analyzing Consistency and Solutions  
  Module 4, Section 4, Lesson 2 Analyzing Consistency and Solutions  

Section 5: Hands-On Practice with Systems  
- Lesson 1: Setting Up and Solving Practice Problems  
  Module 4, Section 5, Lesson 1 Setting Up and Solving Practice Problems  
- Lesson 2: Interpreting the Results Practically  
  Module 4, Section 5, Lesson 2 Interpreting the Results Practically  

Module 5: Practical Applications and Problem Solving

Section 1: Real-World Applications of Linear Algebra  
- Lesson 1: Linear Algebra in Computer Science and Data Analysis  
  Module 5, Section 1, Lesson 1 Linear Algebra in Computer Science and Data Analysis  
- Lesson 2: Applications in Economics and Social Sciences  
  Module 5, Section 1, Lesson 2 Applications in Economics and Social Sciences  

Section 2: Computational Tools and Techniques  
- Lesson 1: Introduction to Software for Linear Algebra  
  Module 5, Section 2, Lesson 1 Introduction to Software for Linear Algebra  
- Lesson 2: Basic Computational Exercises  
  Module 5, Section 2, Lesson 2 Basic Computational Exercises  

Section 3: Hands-On Problem Solving Part I  
- Lesson 1: Step-by-Step Problem Analysis  
  Module 5, Section 3, Lesson 1 Step-by-Step Problem Analysis  
- Lesson 2: Working with Simple Datasets  
  Module 5, Section 3, Lesson 2 Working with Simple Datasets  

Section 4: Hands-On Problem Solving Part II  
- Lesson 1: Developing Your Own Problems  
  Module 5, Section 4, Lesson 1 Developing Your Own Problems  
- Lesson 2: Group Work and Collaborative Solutions  
  Module 5, Section 4, Lesson 2 Group Work and Collaborative Solutions  

Section 5: Review and Course Wrap-Up  
- Lesson 1: Recap of Fundamental Concepts  
  Module 5, Section 5, Lesson 1 Recap of Fundamental Concepts  
- Lesson 2: Final Tips and Pathways for Further Study  
  Module 5, Section 5, Lesson 2 Final Tips and Pathways for Further Study